We here introduce the fidelity for \(|{\psi (t)}\rangle \)  and \(|{\phi (t)}\rangle \) :


F(t) :=|(t)|(t) |2    (0F(t) 1),

where \(\Vert |{a}\rangle \Vert :=\sqrt{\langle a|a\rangle }\)  is the Euclidean norm.
\(F(t)\)  is often employed as a cost function in various quantum science scenario, because it can be easily computed both analytically and numerically [1]}.
If and only if \(|{\psi (t)}\rangle  =e^{i\alpha }|{\phi (t) }\rangle \)  (\(\alpha \in \mathbb {R}\)  is a global phase factor), \(F(t)=1\)  is achieved.
\(F(0)=1\)  and \(F(t)\)  decreases from 1 under the noise.
Therefore, the objective control limit is a fundamental lower bound of \(F(t)\) .
